In a titanic struggle played out over four hours and 43 minutes, world number one Djokovic squandered two match points in the fourth set tiebreak before going on to seal a final clash against either British second seed Andy Murray, the runner-up last year, or 24th seeded Pole Jerzy Janowicz.
The top-seeded Serb, the champion in 2011, leads Murray 11-7 in career meetings but has yet to face Janowicz, the first Polish man to make the semi-finals of a major.
"It was one of the best matches I've been part of, certainly one of the most exciting. It was so close and nothing could separate us," said Djokovic.
"But that's why he is a Grand Slam champion, every time he was in a tough situation he comes up with unbelievable shots.
"I'm very proud to go through. When I lost the fourth set of course it was disappointing because I was close to winning and didn't capitalise. He came up with some big forehands. It was a very high level of tennis.
Del Potro, who had defeated the Serb for the Olympic bronze medal at Wimbledon last year, had matched Djokovic blow for blow in the first set but had to bat back a break point in the sixth game to stay on level terms.
At 6-5, Djokovic stepped it up, a backhand from off his toes landing in the corner before a backhand off-balance from the Argentine sailed long.
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